Manfred Gilli

Technical communique: Understanding complex systems

A system or a model is called complex when it contains such a great number of interconnected variables or elements that it is generally no longer possible to understand its global working without using simplified or condensed forms of the model. With respect to the assumed complexity of a system, simplified models often give rise to some inconsistency and there is a need for guides, or procedures, for a whole understanding of complex interdependent systems which do not imply any loss of information. The proposed approach can also be seen as an extension of the so-called interpretive structural modeling techniques to models embedding many feedbacks between their elements. Briefly summarized, it consists of bringing into focus some particular minimum sets of variables such that, if the variables of one of these sets were fixed by some control, the model would contain no feedbacks. Consequently, it is then possible to define a hierarchical order of the variables of an interdependent system, which is analogous to the hierarchical order given by the reduced graph of a recursive system. An operational computer algorithm is proposed which has been successfully tested with several economic models, some of which contain more than 10 000 circuits.

A global optimization heuristic for estimating agent based models

A continuous global optimization heuristic for a stochastic approximation of an objective function, which itself is not globally convex, is introduced. The objective function arises from the simulation based indirect estimation of the parameters of agent based models of financial markets. The function is continuous in the variables but non-differentiable. Due to Monte Carlo variance, only a stochastic approximation of the objective function is available. The algorithm combines features of the Nelder-Mead simplex algorithm with those of a local search heuristic called threshold accepting. The Monte Carlo variance of the simulation procedure is also explicitly taken into account. We present details of the algorithm and some results of the estimation of the parameters for a specific agent based model of the DM/US-

Applications of optimization heuristics to estimation and modelling problems

foreign exchange market.

The Threshold Accepting Heuristic for Index Tracking

Abstract Estimation and modelling problems as they arise in many fields often turn out to be intractable by standard numerical methods. One way to deal with such a situation consists in simplifying models and procedures. However, the solutions to these simplified problems might not be satisfying. A different approach consists in applying optimization heuristics such as evolutionary algorithms (simulated annealing, threshold accepting), neural networks, genetic algorithms, tabu search, hybrid methods, etc., which have been developed over the last two decades. Although the use of these methods became more standard in several fields of sciences, their use in estimation and modelling in statistics appears to be still limited. A brief introduction to the computational complexity of problems encountered in the fields of statistical modelling and econometrics as well as an overview and classification of the optimization heuristics used is provided. Given the applications presented and the growing availability of optimization heuristics, it is expected that their use will become more frequent in statistics in the near future.

A Global Optimization Heuristic for Portfolio Choice with VaR and Expected Shortfall

We investigate the performance of the threshold accepting heuristic for the index tracking problem. The index tracking problem consists in minimizing the tracking error between a portfolio and a benchmark. The objective is to replicate the performance of a given index upon the condition that the number of stocks allowed in the portfolio is smaller than the number of stocks in the benchmark index. Transaction costs are incurred each time that the portfolio is rebalanced.

Climate Change Impacts on Hydropower Management

Constraints on downside risk, measured by shortfall probability, expected shortfall etc., lead to optimal asset allocations which differ from the mean-variance optimum. The resulting optimization problem can become quite complex as it exhibits multiple local extrema and discontinuities, in particular if constraints restricting the trading variables to integers, constraints on the holding size of assets or on the maximum number of different assets in the portfolio are introduced. In such situations classical optimization methods fail to work efficiently and heuristic optimization techniques can be the only way out. This contribution shows how a particular optimization heuristic, called threshold accepting, can be successfully used to solve complex portfolio choice problems.

Optimal enough

Climate change affects hydropower production by modifying total annual inflow volumes and their seasonal distribution. Moreover, increasing air temperatures impact electricity consumption and, as a consequence, electricity prices. All together, these phenomena may lead to a loss in revenue. We show that an adequate management of hydropower plants mitigates these losses. These results are obtained by resorting to an interdisciplinary approach integrating hydrology, economy and hydropower management in an interdependent quantitative model.

Calibrating the Nelson-Siegel-Svensson Model

An alleged weakness of heuristic optimisation methods is the stochastic character of their solutions: instead of finding the truly optimal solution, they only provide a stochastic approximation of this optimum. In this paper we look into a particular application, portfolio optimisation. We demonstrate that the randomness of the ‘optimal’ solution obtained from the algorithm can be made so small that for all practical purposes it can be neglected. More importantly, we look at the relevance of the remaining uncertainty in the out-of-sample period. The relationship between in-sample fit and out-of-sample performance is not monotonous, but still, we observe that up to a point better solutions in-sample lead to better solutions out-of-sample. Beyond this point there is no more cause for improving the solution any further: any in-sample improvement leads out-of-sample only to financially meaningless improvements and unpredictable changes (noise) in performance.

Heuristic Optimisation in Financial Modelling

The Nelson–Siegel–Svensson model is widely-used for modelling the yield curve, yet many authors have reported ‘numerical difficulties’ when calibrating the model. We argue that the problem is twofold: firstly, the optimisation problem is not convex and has multiple local optima. Hence standard methods that are readily available in statistical packages are not appropriate. We implement and test an optimisation heuristic, Differential Evolution, and show that it is capable of reliably solving the model. Secondly, we also stress that in certain ranges of the parameters, the model is badly conditioned, thus estimated parameters are unstable given small perturbations of the data. We discuss to what extent these difficulties affect applications of the model.

Krylov methods for solving models with forward-looking variables

There is a large number of optimisation problems in theoretical and applied finance that are difficult to solve as they exhibit multiple local optima or are not ‘well-behaved’ in other ways (e.g., discontinuities in the objective function). One way to deal with such problems is to adjust and to simplify them, for instance by dropping constraints, until they can be solved with standard numerical methods. We argue that an alternative approach is the application of optimisation heuristics like Simulated Annealing or Genetic Algorithms. These methods have been shown to be capable of handling non-convex optimisation problems with all kinds of constraints. To motivate the use of such techniques in finance, we present several actual problems where classical methods fail. Next, several well-known heuristic techniques that may be deployed in such cases are described. Since such presentations are quite general, we then describe in some detail how a particular problem, portfolio selection, can be tackled by a particular heuristic method, Threshold Accepting. Finally, the stochastics of the solutions obtained from heuristics are discussed. We show, again for the example from portfolio selection, how this random character of the solutions can be exploited to inform the distribution of computations.